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Constructing and Checking a Turn-Slide Transformation

Constructing and Checking a Turn-Slide Transformation
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2024-12-05 03:36
PolygonAngleVectorTransformationGeometric Construction
In collections Congruence Transformations
This example demonstrates a “turn-slide” (a composite rigid motion) by transforming a polygon through a controlled combination of rotation and translation, showing how the preimage and image are aligned under the transformation. The key ideas are plane isometries and composite transformations: rotate + translate (or an equivalent order), and how corresponding points, segments, and angles match after the motion. The construction uses GeoGebra commands Point, Polygon, Vector, Line, Translate, Mirror, Rotate, Segment, Angle, OrthogonalLine, Circle, and Intersect, building and tracking vertices (A, B, C, D, E, F and their primed images), auxiliary lines/vectors, and the rotation/translation operators that produce the final turn-slide image. It also supports verification via Angle and segment correspondences. Suitable for middle to high school geometry lessons and transformation-focused explorations, including interactive classroom demonstrations.

GeoGebra Commands & Parameters

  1. Point(Circle[A, 1.2851459061133876]) → B
  2. Polygon(A, B, C, D, E, F) → preimage, a_{1}, b_1, c_1, d_1, e, f
  3. Vector(G, H) → Vector
  4. Line(I, J) → lineOfReflection
  5. Translate(A, Vector) → A'_{5}
  6. Translate(B, Vector) → B'_{5}
  7. Translate(C, Vector) → C'_{5}
  8. Translate(D, Vector) → D'_{5}
  9. Translate(E, Vector) → E'_{5}
  10. Translate(F, Vector) → F'_{5}
  11. Polygon(A'_{5}, B'_{5}, C'_{5}, D'_{5}, E'_{5}, F'_{5}) → translated, a', b', c', d', e', f'
  12. Mirror(A, lineOfReflection) → A'_{1}
  13. Mirror(B, lineOfReflection) → B'_{1}
  14. Mirror(C, lineOfReflection) → C'_{1}
  15. Mirror(D, lineOfReflection) → D'_{1}
  16. Mirror(E, lineOfReflection) → E'_{1}
  17. Mirror(F, lineOfReflection) → F'_{1}
  18. Rotate(A, dummyRotationAngle, Center) → A'_{2}
  19. Rotate(B, dummyRotationAngle, Center) → B'_{2}
  20. Rotate(C, dummyRotationAngle, Center) → C'_{2}
  21. Rotate(D, dummyRotationAngle, Center) → D'_{2}
  22. Rotate(E, dummyRotationAngle, Center) → E'_{2}
  23. Rotate(F, dummyRotationAngle, Center) → F'_{2}
  24. Polygon(A'_{2}, B'_{2}, C'_{2}, D'_{2}, E'_{2}, F'_{2}) → rotated, a'_{2}, b'_{2}, c'_{2}, d'_{2}, e'_{2}, f'_{2}
  25. Line(K, lineOfReflection) → h
  26. Point(h) → L
  27. Vector(K, L) → Vector2
  28. Mirror(A, lineOfReflection) → A'_{3}
  29. Mirror(B, lineOfReflection) → B'_{3}
  30. Mirror(C, lineOfReflection) → C'_{3}
  31. Mirror(D, lineOfReflection) → D'_{3}
  32. Mirror(E, lineOfReflection) → E'_{3}
  33. Mirror(F, lineOfReflection) → F'_{3}
  34. Polygon(A'_{3}, B'_{3}, C'_{3}, D'_{3}, E'_{3}, F'_{3}) → intermediary, a'_{3}, b'_{3}, c'_{3}, d'_{3}, e'_{3}, f'_{3}
  35. Translate(A'_{3}, Vector2) → M
  36. Translate(B'_{3}, Vector2) → N
  37. Translate(C'_{3}, Vector2) → O
  38. Translate(D'_{3}, Vector2) → P
  39. Translate(E'_{3}, Vector2) → Q
  40. Translate(F'_{3}, Vector2) → R
  41. Polygon(M, N, O, P, Q, R) → image_{1}, m, n, o, p, q, r
  42. Rotate(preimageList, angleAnimator, rotationCenter) → RTanimRotList
  43. Rotate(A, rotationAngle) → A'_{4}
  44. Rotate(C, rotationAngle) → C'_{4}
  45. Rotate(D, rotationAngle) → D'_{4}
  46. Rotate(E, rotationAngle) → E'_{4}
  47. Rotate(F, rotationAngle) → F'_{4}
  48. Rotate(A, rotationAngle, rotationCenter) → A'_{6}
  49. Rotate(C, rotationAngle, rotationCenter) → C'_{6}
  50. Rotate(D, rotationAngle, rotationCenter) → D'_{6}
  51. Rotate(E, rotationAngle, rotationCenter) → E'_{6}
  52. Rotate(F, rotationAngle, rotationCenter) → F'_{6}
  53. Rotate(preimageList, rotationAngle, rotationCenter) → RTintermList
  54. Polygon(RTanimRotList) → RTanimRot
  55. Polygon(RTintermList) → RTinterm
  56. Segment(rotationCenter, RTanimRotList(2)) → RFrotSegment
  57. Line(rotationCenter, V) → a
  58. Rotate(V, rotationAngle / 2, rotationCenter) → V'
  59. Angle(V, rotationCenter, V') → linesAngle
  60. Line(rotationCenter, V') → b
  61. Vector(S, T) → translVector
  62. OrthogonalLine(W, translVector, xOyPlane) → c
  63. Circle(W, abs(translVector) / 2) → k
  64. OrthogonalLine(W, c, xOyPlane) → i
  65. Circle(W, abs(translVector) / 2) → s
  66. Intersect(i, s, 2) → Z
  67. Line(Z, c) → d
  68. Translate(RTinterm, translVector) → image
  69. Mirror(A, lineOfReflection) → A'
  70. Mirror(B, lineOfReflection) → B'
  71. Mirror(C, lineOfReflection) → C'
  72. Mirror(D, lineOfReflection) → D'
  73. Mirror(E, lineOfReflection) → E'
  74. Mirror(F, lineOfReflection) → F'
  75. Polygon(A', B', C', D', E', F') → preimage', l, t, a_{2}, f_1, g_1, h_1
  76. Polygon(RTanimTranslList) → RTanimTransl
  77. Polygon(TRanimTranslList) → TRanimTransl
  78. Polygon(TRintermList) → TRinterm
  79. Rotate(TRintermList, angleAnimator, rotationCenter) → TRanimRotList
  80. Polygon(TRanimRotList) → TRanimRot
  81. Segment(rotationCenter, TRanimRotList(1)) → rotSegment
  82. Rotate(TRinterm, rotationAngle, rotationCenter) → TRinterm'
  83. Translate(A, translVector) → A'_{7}
  84. Translate(C, translVector) → C'_{7}
  85. Translate(D, translVector) → D'_{7}
  86. Translate(E, translVector) → E'_{7}
  87. Translate(F, translVector) → F'_{7}
  88. Mirror(A, c) → A'_{8}
  89. Mirror(B, c) → B'_{4}
  90. Mirror(C, c) → C'_{8}
  91. Mirror(D, c) → D'_{8}
  92. Mirror(E, c) → E'_{8}
  93. Mirror(F, c) → F'_{8}
  94. Polygon(A'_{8}, B'_{4}, C'_{8}, D'_{8}, E'_{8}, F'_{8}) → preimage'_{1}, a'_{8}, b'_{4}, c'_{8}, d'_{8}, e'_{8}, f'_{8}
  95. Mirror(preimage'_{1}, d) → poly1
  96. Mirror(poly1, a) → poly1'
  97. Mirror(poly1', b) → poly1''
  98. Mirror(A, a) → A'_{9}
  99. Mirror(B, a) → B'_{6}
  100. Mirror(C, a) → C'_{9}
  101. Mirror(D, a) → D'_{9}
  102. Mirror(E, a) → E'_{9}
  103. Mirror(F, a) → F'_{9}
  104. Polygon(A'_{9}, B'_{6}, C'_{9}, D'_{9}, E'_{9}, F'_{9}) → preimage'_{2}, a'_{9}, b'_{6}, c'_{9}, d'_{9}, e'_{9}, f'_{9}
  105. Mirror(A'_{9}, b) → B_1
  106. Mirror(B'_{6}, b) → C_1
  107. Mirror(C'_{9}, b) → D_1
  108. Mirror(D'_{9}, b) → E_1
  109. Mirror(E'_{9}, b) → F_1
  110. Mirror(F'_{9}, b) → G_1
  111. Polygon(B_1, C_1, D_1, E_1, F_1, G_1) → poly2, g, j, d_{2}, e_1, i_1, j_1
  112. Mirror(B_1, c) → H_1
  113. Mirror(C_1, c) → I_1
  114. Mirror(D_1, c) → J_1
  115. Mirror(E_1, c) → K_1
  116. Mirror(F_1, c) → L_1
  117. Mirror(G_1, c) → M_1
  118. Polygon(H_1, I_1, J_1, K_1, L_1, M_1) → poly2', k_1, l_1, m_1, n_1, p_1, q_1
  119. Mirror(poly2', d) → poly2''

AI Drawing Prompt Reference

Draw a polygon as the original shape, then add a line as the reflection axis and a vector for translation. First translate the polygon along the vector to get a translated copy, then reflect it across the axis to get a mirrored copy. Next, rotate the original polygon about a center point by an angle to get a rotated copy. Finally, reflect the original polygon across the axis and then translate the reflected result by another vector to obtain the final composite image. Use dashed lines or different colors to mark corresponding points, sides, and angles, illustrating the combined rotation and translation rigid motion.
The prompt above was compiled from the original drawing commands and is for reference only.